Woohoo, Hungarian names! I can't really say anything more than Chrisell, except about Buda.
1.
Alex: as a full formal name,
Alex is a male name. However, an
Alexandra can also have the informal nn
Alex.
2. ADél is female, the Hun. form of
Adele.
3. Buda is male: according to a myth he was the elder brother of
Attila the Hun (as you may now, it's a popular belief - though not true - that the Hungarians are related to the Huns). The reason why you found it as a place name is because our capital, Budapest, was made from the uniting of two small towns, Buda and Pest it the 1800s.
Hungary is a conservative country when it comes to naming: we have no unisex names...
I hope my little lecture was not too boring :-)